The Codifference Function of Moving Average Process with Symmetric Alfa-Stable Innovation

$S\alpha S$ The research entitled “The Codifference Function of Moving Average Processes With Symmetric $\alpha$ -stable Innovation ” was conducted by Iqbal Kharisudin under the guidance of Dr.rer.nat., Dedi Rosadi, M.Sc., Dr. Abdurakhman, M.Si., and Dr. Suhartono, M.Sc. in 2018.

The following is the abstract of this research.

ABSTRACT

Most of statistical models require the existence of a second-order moment or based on the distribution with finite variance. We can study the dependency structure of the model based on second-order moment. In the time series modeling with finite variance innovation (eg Gaussian model), the autocorrelation function (ACF) plays an important role. One of the applications of ACF on Box-Jenkins time series modeling is for model identification. In the MA( $q$ ) models, we found that the ACF values is zero after lag q. Such essential properties are used as a basis for identifying the MA process given by time-series data.

In the stable time series modeling, ACF function can not be used, because it requires finite variance ( $\sigma <\infty$ ). Therefore, in the modeling of time series with stable assumptions we use alternative dependency measures, such as coddifference function or covariance function. The covariance function and its application were studied in Gallagher (2000, 2001). Rosadi and Deistler (2011) discussed the generalized form of the sample codifference functions for linear stationary models and its asymptotic property. The special case of iid process has also been discussed. Based on the literature review and the author’s knowledge, the study of samples codifference function and its asymptotic properties for MA( $q$ ) model with symmetric $\alpha$ -stable ( $S\alpha S$ ) innovation are still an open problem.

In this research, we examine the properties of codifference function, its estimator, and the asymptotic properties of sample codifference function for MA( $q$ ) $S\alpha S$ . We can define the sample autocodifference (ACodF) by using sample normalized codifference function. In this study, we examine the computational simulation and application of sample autocodifference function (ACodF) for the identification of MA( $q$ ) process with Gaussian ( $\alpha = 2$ ) and $S\alpha S$ non Gaussian ( $\alpha < 2$ ) innovation using asymptotic confidence intervals.

The main results of this research are the asymptotically normal distributed properties of the sample autocodifference function (ACodF, $\hat{I}(k)$ ) of MA( $q$ ) $S\alpha S$ process for $k > q.$ We found that the sample ACodF for MA( $q$ ) $S\alpha S$ process was asymptotically normal distributed with covariance matrix depends on its coefficients. These results are expressed in the theorems and corollary. The obtained results was an extension of the asymptotic property of the sample ACodF for the iid case. We can do inference about ACodF for MA( $q$ ) $S\alpha S$ process based on these properties. For application purposes, we proposed an iterative procedure for the $q$ order identification of MA( $q$ ) $S\alpha S$ process.

A simulation study was conducted to evaluate computationally the asymptotic properties of the sample ACodF. From the simulation, we found that the order identification of the MA(1) and MA(2) processes using the corrected confidence interval bounds $\hat{I}$ of the ACodF can improve the identification performance of the desired order. The model identification increases better than using ACF $\hat{\ro}$ , auto-covariation (ACovF) $\hat{\lambda}$ , or using ACodF by MA(0) confidence interval bound, especially if the sample size is large enough (no less than 100). Based on simulation study, we obtain these improvements are about $59,5%-86,8%$ . These results are consistent with the asymptotic properties of sample ACodF for MA( $q$ ) $S\alpha S$ process. In the last section, we discuss the application of the sample ACodF for MA model identification base on iterative procedure for returns of stock data in Indonesia.

Keywords: dependence measure, infinite variance, symmetric $\alpha$ -stable, MA( $q$ )